Question

Your friend claims that because each midsegment is half as long as the corresponding side of the triangle, the perimeter of the midsegment triangle is half the perimeter of the original triangle. Is your friend correct ? Explain your reasoning.

Answer 1

The claim made by your friend is incorrect. The perimeter of the midsegment triangle is not half the perimeter of the original triangle. To understand why, let's consider an example.

Step-by-step explanation:

The claim made by your friend is incorrect. The perimeter of the midsegment triangle is not half the perimeter of the original triangle. To understand why, let's consider an example. Suppose we have a triangle with sides measuring 4, 5, and 6 units. The midsegments will be half the length of the corresponding sides, so they will measure 2, 2.5, and 3 units.

The perimeter of the original triangle is 4 + 5 + 6 = 15 units, while the perimeter of the midsegment triangle is 2 + 2.5 + 3 = 7.5 units. As can be seen, the perimeter of the midsegment triangle is not half the perimeter of the original triangle. Therefore, your friend's claim is incorrect.

Answer 2

Answer:
The claim is correct

Step-by-step explanation:
Assume the given triangle ABC
perimeter of triangle ABC = AB + BC + CA ............> I

Now, we have:
D is the midpoint of AB, this means that:
AD = DB = (1/2) AB ..........> 1
E is the midpoint of AC, this means that:
AE = EC = (1/2) AC ...........> 2
DE is the midsegment in triangle ABC, this means that:
DE = (1/2) BC ...........> 3
perimeter of triangle ADE = AD + DE + EA
Substitute in this equation with the corresponding lengths in 1,2 and 3:
perimeter of triangle ADE = (1/2) AB + (1/2) BC = (1/2) AC
perimeter of triangle ADE = (1/2)(AB+BC+AC) .........> II

From I and II, we can prove that:
perimeter of triangle ADE = (1/2) perimeter of triangle ABC
Which means that:
perimeter of midsegment triangle is half the perimeter of the original triangle.

Hope this helps :)